M=5 (3,7) runlength limited code for multi-level data

ABSTRACT

A system for encoding digital data with an M-ary (d,k) code to provide multi-level coded data where M&gt;2 comprises an M-ary (d,k) encoder for accepting digital input data. The M-ary (d,k) coder encodes the digital input data to produce a plurality of code symbols, where each code symbol is at one of M levels and each pair of non-zero code symbols is separated by at least d but no more than k zeros. A waveform encoder converts the code symbols into waveform signal amplitudes compatible with a multi-level channel. Each waveform signal amplitude is generated by modulo M addition of a current code symbol with a previous waveform signal amplitude. A specific M=5 (3,7) code provides high density recording to a multi-level storage medium. The M=5 (3,7) encoder is implemented using a six-state encoder and a modulo five waveform encoder.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to multi-level codingtechniques, and more specifically to a system and method for coding datausing an M=5 (3,7) runlength limited code, such code being particularlyuseful for storing and/or transmitting multi-level data.

2. Related Art

Various techniques are currently in use for increasing the recordingdensity on various digital recording mediums such as magnetic tapes anddisks and in other similar media. One technique utilizes an approachknown as runlength-limited coding. This technique requires that each "1"in a coded bit sequence must be separated by a specified number of"0's". This number of zeros must be at least a minimum quantity, d, toreduce intersymbol interference, and is limited to a maximum quantity kfor self clocking purposes. Such codes are generally referred to as(d,k) run-length-limited (RLL) codes.

Because conventional recording techniques use saturation recording tostore information, binary recording techniques are often used to markthe recording medium. As a result, conventional (d,k) RLL codesdeveloped to date are developed and optimized to store binary data.

Significant advances in data storage materials have lead to thedevelopment of a storage medium that provides a linear responsecharacteristic. One such material providing a linear responsecharacteristic is an electron trapping material such as is disclosed inU.S. Pat. Nos. 4,864,536, 5,007,037, and 5,142,493, all to Lindmayer.

Using a material which provides a linear response characteristic yieldsan advantage over saturation-type media in that it adds an analogdimension to the storage capacity of the medium. Because the response islinear, the linear-response storage materials provide the ability toencode information in two dimensions--amplitude and phase.

As a result, the storage medium is no longer confined to storing binaryor even tri-level data. Instead, the concept of M-ary, or non-binary,data coding and storage is provided. The increased symbol set allowed bysuch encoding provides the opportunity to dramatically increase the datarecording density and transfer rate of the storage device. For example,the potential storage capacity of a single 51/4 inch disk can beextended to several gigabytes if that disk is implemented using electrontrapping materials with M-ary (multi-level) data coding.

SUMMARY OF THE INVENTION

The present invention is directed toward an M=5 (3,7) runlength-limitedcode for multi-level data storage and/or communications. Also disclosedherein is a system and method for generating multi-level data from abinary input data stream using an M-ary (d,k) code such as the M=5 (3,7)runlength limited code.

According to the invention input data bits are coded using an M-ary(d,k) encoder to generate code symbols. Specifically, input data bits xare encoded using the M-ary (d,k) encoder to generate code symbols y,where:

x=(x₀,x₁, . . . x_(p-1)), for p input data bit

y=(y₀,y₁, . . . y_(pq-1)), where pq=the code rate R

The code symbols y are multi-level symbols, each being at one of Mlevels. The code symbol stream provided meets the code specifications ofno fewer than d and no greater than k zeros between each non-zerosymbol. Thus, according to the M=5 (3,7) code disclosed herein, codesymbols are each at one of five levels and there is a minimum of threeand a maximum of seven zeros between each non-zero symbol.

The code symbols y are encoded to generate a series of waveformamplitudes z. The waveform amplitudes are generated by modulo M additionof the current code symbol y_(i) with the previous waveform amplitudez_(i-1). The waveform amplitudes are used to modulate a write laser tostore information onto an optical disk or to otherwise store orcommunicate the input data bits x in the form of multi-levelinformation.

To recover the original data bits x from the medium, the informationstored on the medium is read and the resultant signal amplitudes aredecoded. Two stages of decoding are provided. In the first stage, thewaveform amplitudes are decoded to generate code symbols. In the secondstage, the code symbols are decoded to recover the original data bits.

The system and method described herein for coding and decoding the datais described with respect to the M=5 (3,7) code disclosed herein.However, while other codes may not be obvious in light of the codedisclosed herein, given a specific M-ary (d,k) code, after reading thebelow disclosure, it will become apparent to a person skilled in therelevant art how the disclosed system and method can be implemented tocode and decode the data using such a specific M-ary (d,k) code.

An advantage of the invention is that the recording density can beincreased by coding the input data using the disclosed M=5 (3,7) code.Because there are at least d zeros between every non-zero code symbol,there can be at least d+1 symbols stored for a given minimum featuresize. Given a code of rate R, the density achievable is (d+1)R.

It should be noted that the coding techniques described herein are notlimited to optical recording. Other recording systems capable ofsupporting multi-level data would benefit from the coding techniquesdescribed herein. Additionally, these coding techniques are alsoapplicable to communications systems where the communications channel iscapable of supporting multi-level data. In such an implementation, thewaveform amplitudes could be used to modulate a signal carrier fortransmission across the multi-level channel.

Further features and advantages of the present invention, as well as thestructure and operation of various embodiments of the present invention,are described in detail below with reference to the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is described with reference to the accompanyingdrawings. In the drawings, like reference numbers indicate identical orfunctionally similar elements. Additionally, the left-most digit(s) of areference number identifies the drawing in which the reference numberfirst appears.

FIG. 1 is a diagram illustrating a portion of an optical storage mediumand features thereon.

FIG. 2 is a diagram illustrating a data storage system for storing amulti-level coded data.

FIG. 3 is a diagram illustrating an encoder for encoding multi-leveldata using a multi-level (d,k) code.

FIG. 4 is a flow diagram illustrating the operation of a multi-stateencoder.

FIG. 5 is a diagram illustrating an example input data stream beingcoded using the encoder illustrated in FIG. 4.

FIG. 6 is a diagram illustrating a decoder for decoding multi-level(d,k) coded data.

FIG. 7 is a diagram illustrating a sliding block decoder.

DETAILED DESCRIPTION OF THE EMBODIMENTS

1. Introduction

The present invention is directed toward an M-ary (d,k) code for use instorage and/or transmission of multi-level data, and a system and methodfor generating the same. This discussion is provided in severalsections. In Section 2, before discussing the invention in detail, abrief discussion of binary (d,k) codes is presented. In Section 3, acoding system for coding input data using an M-ary (d,k) code isdescribed. In Section 4, some code design factors are described. InSection 5, a specific M=5 (3,7) code is described that is particularlyuseful for storage and/or transmission of multi-level data. Section 5also provides a detailed description of the encoder used to code datausing the disclosed M=5 (3,7) code. In Section 6 there is presented adiscussion of a sliding block decoder generally useful for decodingM-ary (d,k) codes, as well as a specific decoder for decoding the M=5(3,7) code described in Section 4.

The present invention is generally described in terms of an embodimentwhere multi-level data are stored on and retrieved from an optical disk.In such an embodiment, the encoder and decoder described herein may beimplemented for multi-level data storage in an optical disk system suchas that disclosed in U.S. Pat. No. 5,007,037, which is incorporatedherein by reference. However, after reading the below description, itwill become apparent to a person skilled in the relevant art how thesystems and methods of the invention can be implemented for alternativeapplications such as multi-level data communications and storage ofmulti-level data on media other than optical disks. In yet otheralternative embodiments, the invention can be implemented to transmitand receive data across an M-ary communications channel. In thisdocument, the term "channel" is sometimes used to refer to the opticaldisk. It should be understood that in the alternative embodiments theterm channel can be used to describe alternative storage media and/orcommunications channels.

Additionally, after reading the below detailed description of theencoder and decoder of the present invention, it will become apparent toa person skilled in the art that these disclosed components can beimplemented using hardware, software or a combination of both hardwareand software.

2. Binary (d,k) Run-Length-Limited Coding

Using conventional binary storage techniques, data are written to astorage medium using binary 1's and 0's. Using run-length limitedcoding, the data to be recorded are first encoded using arun-length-limited encoder. As stated above, with a (d,k) RLL code,there is a minimum and a maximum number of 0's that can occur betweeneach pair of 1's, as specified by the code parameters d and k. Thus, the(d,k) RLL encoder results in a 1 followed by at least d and at most k0's before another 1 occurs. Such (d,k) RLL codes for binaryapplications are well known in the art.

To write input data to the saturation storage medium, the input data arefirst encoded to create symbols. The encoder is designed such that theoutput symbols conform to the code specifications: each 1 is followed bya minimum of d and a maximum of k 0's. These symbols are then written tothe storage medium as a series of features. For example, for opticaldisks, a feature is the presence or absence of a burn mark. In amagnetic tape, the feature is a grouping of magnetic particles orientedin a particular manner.

FIG. 1 illustrates a portion of an optical storage medium 108 thatstores binary data as features 102. Illustrated are shaded and unshadedfeatures 102, indicating the absence or presence of a burn mark. Alsoillustrated in FIG. 1 is an example symbol stream 112 that is used towrite the features 102 to medium 108. As discussed above, symbol stream112 is the result of encoding input data using a (d,k) RLL code. Symbolstream 112 is written to medium 108 using an additional coding step asfollows: the occurrence of a 1 indicates a change in the type of mark(i.e., absence or presence of a bum mark); the occurrence of a 0indicates no change in mark type. This additional coding step issometimes referred to as non-return-to-zero interleaved (NRZI)precoding.

Note that with a (d,k) RLL code where d=2, the minimum number of symbolsrepresented by a single feature 102 is three. Thus, for a minimumfeature size (e.g., for the smallest spot size on medium 108) threesymbols can be represented. As a result, for a rate 1/2 code, where eachinput bit corresponds to two symbols, the minimum density of datawritten to medium 108 is 1.5 bits per feature. Thus, as a result ofusing the (d,k) RLL coding in the described scenario, the density of therecording system can be increased by 50%.

3. A Generic M-ary Runlength-Limited (d,k) Coder

A few advantages of (d,k) coding are described above with reference tobinary data systems. It is important to note that similar gains indensity can be realized for storage and communications systems usingmulti-level (versus binary) data. To optimize such gains, however, newcodes and new coding techniques are required to take full advantage ofthe unique properties of the multi-level data. Thus, the inventors havedeveloped a system and method for encoding data using M-ary (d,k) codes.

FIG. 2 is a high-level block diagram generally illustrating amulti-level data recording system 200. The recording system is comprisedof an encoder 204, multi-level storage medium 208 and a decoder 212. Asstated above, multi-level storage medium 208 can be replaced with acommunications channel allowing the multi-level (d,k) code and codingscheme to be used with multi-level data communications. In such acommunications embodiment, encoder 204 and decoder 212 can be providedon both ends of the communications channel to allow hi-directional codedcommunications.

According to the invention, input data bits x=(x₀, x₁, . . . , x_(p-1))are the actual data that a user wishes to be stored on and retrievedfrom medium 208. However, to take full advantage of the multi-levelproperties of medium 208 and to increase the storage density achieved,the invention provides for the encoding of input data x so thatmulti-level data is stored on medium 208. Such encoding is accomplishedusing encoder 204.

In one embodiment, encoder 204 is implemented as a two-stage encoder.FIG. 3 is a block diagram illustrating encoder 204 in this embodiment.Referring to FIG. 3, encoder 204 is comprised of a symbol encoder 304and a waveform encoder 308, referred to herein as encoder 304 andencoder 308, respectively.

Encoder 304 is a (d,k) RLL symbol encoder that accepts the input databits x and converts them to a stream of code symbols y=(y₀, y₁, . . . ,y_(q-1)). To take advantage of the multi-level characteristics of medium208, encoder 304 generates code symbols y at M different levels, whereM>2. A second feature of encoder 304 is that the stream of code symbolsy meets the code specification that there are no fewer than d and nogreater than k zeros between each non-zero code symbol. Another featureof encoder 304 is that there are R input bits x_(i) for each code symboly_(i) generated. This feature is referred to as the code rate. Thus fora rate 1/N code (i.e., where R=1/N), there are N code symbols y_(i)generated for each input data bit x_(i).

Waveform encoder 308 accepts code symbols y and converts them to aseries of waveforms z=(z₀, z₁, . . . , z_(q-1)), each of a givenamplitude. Waveform encoder 308 generates output waveforms z, each atone of M different amplitudes. The waveform amplitudes are generated bymodulo M addition of the current code symbol y_(i) with the previouswaveform amplitude z_(i-1). It is the waveforms z which are used tomodulate a write laser for storing data onto medium 208 (or transmittedacross the multi-amplitude channel).

Because there are at a minimum d zeros, between each non-zero codesymbol, each waveform amplitude z generated represents, at a minimum,d+1 code symbols y. Also, because them are 1/R code symbols y generatedfor each input data bit x, each waveform amplitude z generatedrepresents, at a minimum, (d+1)R data bits x. Thus, for a minimumfeature size (e.g., the smallest spot that can be written to the storagemedium or the shortest pulse that can be transmitted across thecommunications channel), the density achieved is D=(d+1)R.

Consider this density in terms of the specific M=5 (3,7) code describedbelow. The rate of this code is R=3/4 and d=3. For this code, thedensity is 3.0 input data bits xi per minimum feature size. In otherwords, 3.0 input data bits can be stored using a single minimum-sizefeature.

4. Code Design Factors

The coding system and method described above can be used to code inputdata using any of number of M-ary (d,k) codes. However, the design of aspecific M-ary (d,k) code is not a trivial matter.

There are several factors that must be taken into consideration whendesigning M-ary (d,k) codes. The specific coding scheme used toimplement a given code impacts the performance of the encoding anddecoding systems. For example, the code should be designed such thatencoders and decoders can be implemented in a straight-forward andcost-effective manner. Toward this end, the desirable code can beimplemented using encoders that have a relatively small number of stateswhile still meeting the code parameters (i.e., M, d and k).Additionally, the decoder required to decode the coded data should beimplementable using a small sliding block window size and the decodershould have a relatively a small table size.

Further, the code design must result in an encoder and decoder thatoperate such that errors in the decoder side have a minimum effect. Itwould be undesirable for errors to propagate too far in the decoded datastream. In some less-than-desirable systems, a single decoding error canresult in a very large, if not infinite, number of additional decodingerrors.

Achievement of such design goals is not a trivial matter. Not everyM-ary (d,k) code of a given set of parameter values will perform in adesirable manner. Provided in the next section of this document is aspecific M=5 (3,7) runlength-limited code that meets the design goalsoutlined above.

5. M=5 (3,7) Runlength-Limited Code

As stated above, code design is not a trivial matter as there areseveral coding factors that affect the performance of the system. Thespecific code implemented for a given set of parameters can impact theoverall performance of the system. This section of the patent documentdescribes a specific M-ary (d,k) code and the specific encoder used toimplement this code. In particular, this section describes a specificM=5 (3,7) runlength-limited code having a rate R=3/4.

Table 1 is a state table illustrating a state machine 400 for encodinginput data x to generate code symbols y for the M=5 (3,7) code (i.e. aspecific version of encoder 304). State machine 400 is a six-state, M=5(3,7) encoder. For each input data bit set x_(i) . . . x_(i+2), anoutput code symbol set y_(i) . . . y_(i+3) is produced; each output codesymbol y_(i) is at one of five levels (0-4). The various states as wellas the inputs and outputs of state machine 400 are illustrated usingTable 1.

FIG. 4 is an operational flow diagram illustrating the operation ofstate machine 400. The operation of state machine 400 is now describedwith reference to Table 1 and FIG. 4. In a step 404, the operationstarts in a given state at time t=0. In a step 408, an input data bitset x_(i) . . . x_(i+2) is received. In a step 412, Table 1 is used todetermine the output code symbol set y_(i) . . . y_(i+3) and next statefrom the current state and the input data bit set x_(i) . . . x_(i+2)received in step 408. In a step 416, encoder 304 outputs code symbol sety_(i) . . . y_(i+3) determined in step 412. In a step 420, state machine400 transitions to the next state determined in step 412. At this time,the operation continues at step 408 as indicated by flow line 426.

As is evident by the above description, an output code symbol set y_(i). . . y_(i+3) of encoder 304 is a function of the input data bit setx_(i) . . . x_(i+2), and the current state 404 of encoder 304. For theM=5 (3,7) encoder of the present invention, Table 1 describes the outputcode symbol set y_(i) . . . y_(i+3) and next state of encoder 304 foreach current state and input data bit set x_(i) . . . x_(i+2).

                  TABLE 1                                                         ______________________________________                                        INPUT         OUTPUT   NEXT STATE                                             ______________________________________                                        STATE 0                                                                       000           0000     2                                                      001           0000     3                                                      010           0000     4                                                      011           0001     0                                                      100           0000     5                                                      101           0002     0                                                      110           0003     0                                                      111           0004     0                                                      STATE 1                                                                       000           0010     0                                                      001           0010     1                                                      010           0030     0                                                      011           0030     1                                                      100           0020     0                                                      101           0020     1                                                      110           0040     0                                                      111           0040     1                                                      STATE 2                                                                       000           0100     2                                                      001           0100     0                                                      010           0100     1                                                      011           0300     2                                                      100           0100     3                                                      101           0300     0                                                      110           0300     1                                                      111           0300     3                                                      STATE 3                                                                       000           0200     2                                                      001           0200     0                                                      010           0200     1                                                      011           0400     2                                                      100           0200     3                                                      101           0400     0                                                      110           0400     1                                                      111           0400     3                                                      STATE 4                                                                       000           2000     2                                                      001           2000     3                                                      010           2000     4                                                      011           4000     5                                                      100           2000     5                                                      101           4000     4                                                      110           4000     3                                                      111           4000     2                                                      STATE 5                                                                       000           1000     2                                                      001           1000     3                                                      010           1000     4                                                      011           3000     5                                                      100           1000     5                                                      101           3000     4                                                      110           3000     3                                                      111           3000     2                                                      ______________________________________                                    

To further illustrate the operation of symbol encoder 304, consider anexample input data stream x as illustrated in Table 2. In this example,assume the encoder 304 starts in state 5 and that the first input databit set x_(i) . . . x_(i+2) is `101`. Referring to Table 1 for state 5,the row where x_(i) . . . x_(i+2) =`101` indicates that encoder 304outputs symbol `3000` and transitions to state 4.

FIG. 5 is a diagram further illustrating the example. FIG. 5 illustratesa current state 508 and a new state 512 for the example input datastream provided in Table 2. Referring to FIG. 5 and state machine 400,given the input data stream illustrated in Table 2, for each input databit set x_(i) . . . x_(i+2), encoder 304 transitions from a currentstate 508 to a new state 512 and produces an output symbol set y_(i) . .. y_(i+3). FIG. 5 illustrates the example for 12 data bits x of theinput data stream while Table 2 is an example for 15 data bits x.

                  TABLE 2                                                         ______________________________________                                        x     1  0  1   1  0  1 1  0  1 0  1  0                                                                             1  0  1                                 y     300       040     004     000   200                                     z     333       322     221     111   333                                     ______________________________________                                    

As described above with reference to FIG. 3, code symbols y are furthercoded by waveform encoder 308 to generate a sequence of amplitudes zused to modulate the write laser. According to one embodiment, waveformencoder 308 is a modulo five encoder that, for a given time, determinesthe waveform amplitude z_(i) by

    z.sub.i =(z.sub.i-1 +y.sub.i) mod M

Note that z is a transformed version of y, where the difference (mod M)between waveform amplitudes z_(i) and z_(i-1) is the coded symbol y_(i).Thus, in this embodiment, each new code symbol y_(i) is modulo fiveadded to the previous waveform amplitude z_(i-1) to result in thecurrent waveform amplitude z_(i). In following the example describedabove and illustrated in FIG. 5 and Table 2, the code symbols y areencoded by waveform encoder 308 to waveform amplitudes z, as illustratedin Table 2.

To briefly summarize the encoding process, input digital data bits x arefirst encoded using an M=5 (3,7) symbol encoder to produce code symbolsy at M=5 levels. Code symbols y are then encoded by waveform encoder 308to produce waveform amplitudes z. According to the current code, M=5amplitudes (illustrated in the above examples as 0-4) are possible.These amplitudes are written to the media by the write laser asillustrated in FIG. 3.

6. Decoder

In order to recover the original data bits x from the amplitudes zstored on the media (or transmitted across the communications channel) adecoder is implemented. In one embodiment, the decoder uses a stateindependent lookup table to decode amplitude levels z read from thedisk. In this embodiment, the decoder is a sliding block decoder, wherea sliding window is used to decode amplitudes read from the disk.

FIG. 6 is a block diagram illustrating the decoder. As illustrated inFIG. 6, decoder 212 is comprised of a waveform decoder 604 and a slidingblock decoder 608. Waveform decoder 604 receives the signal amplitudesz' (multi-level) from the media and converts them to a sequence ofdigital code symbols y'. Sliding block decoder 608 performs a tablelookup to convert output code symbols y' into output data bits x'. Notethat in an error free environment, output code symbols y' correspondidentically to input code symbols y and output data bits x' correspondidentically to input data bits x.

Any of a number of different techniques can be implemented to readwaveform amplitudes z off of the media to provide the signal amplitudesz' to waveform decoder 604. It should be noted that the actual readscheme implemented is not important to the invention as long as data areread accurately.

As stated above, the purpose of waveform decoder 604 is to decode signalamplitudes z' to recover the actual code symbols y'. Waveform decoderconverts a sequence of amplitudes z'=(z'₀, z'₁, . . . , z'_(N-1)) wherez'_(i) .di-elect cons. 0,A!, for some real number A, to a sequence ofcode symbols y'=(y'₀, y'₁, . . . y'_(N-1)).

The operation of waveform decoder 604 is now described. For a given timei, the code symbol y'_(i) is determined by

    y'.sub.i =(z'.sub.i -z'.sub.i-1) mod M

According to this technique, each output symbol y' is determined asbeing the modulo M difference between the current amplitude waveformz'_(i) and the previous amplitude waveform z'_(i-1). This decodingessentially undoes the coding performed by waveform encoder 308.Specifically, for M=5 (d,k) codes, such as the M=5 (3,7) code describedabove in Section 5, the decoding is implemented using a modulo fivedifference.

6.1 Sliding Block Decoder

The purpose of sliding block decoder 608 is to convert the sequence ofcode symbols y' into a sequence of data bits x' that coincide with inputdata bits x. In a preferred embodiment, the decoder is a stateindependent lookup table. The size of the lookup table is dependent onthe specific code implemented. Therefore, the design of encoder 304affects the size of the lookup table implemented.

Sliding block decoder 608 is illustrated in FIG. 7. Sliding blockdecoder comprises a sliding window 704 to decode the code symbols y' toarrive at output data bits x'. Whereas encoder 304 accepts one data bitset x_(i) . . . x_(i+2) and generates one code symbol set y_(i) . . .y_(i+3) therefrom, sliding block decoder 608 must look at multiple codesymbols y' to decode one data bit set. Specifically, for the encoderdescribed above with reference to Table 1, sliding block decoder 608requires a block of twelve code symbols y_(i) ' to uniquely decode onedata bit set x_(i) ' . . . x_(i+2) '.

The actual size of sliding window 704 is determined during code design.An optimally sized sliding window 704 is large enough such that thesymbols y' within sliding window 704 unambiguously define the correctoutput bit set x' without being so large that unneeded code symbols y'are included therein. In other words, the window size is chosen as thesmallest window which guarantees unique decodability.

With the use of a sliding window 704 to decode the data, knowledge ofstate information is not required. In fact, the contents of slidingwindow 704 at any given time contain sufficient information such thatstate information can be determined therefrom. Thus, sliding blockdecoder 608 is computationally less complex than conventional decoders;more importantly, the sliding block decoder limits the propagation oferrors.

Sliding window 704 actually represents the past, present and future. Thecode symbols y' actually being decoded to generate data bits x'represent the present. Past symbols y' are those occurring earlier intime than the present symbols. Likewise, future symbols y' are thosethat occur after the present symbols. In FIG. 7, the past is representedby `3000`, the present by `4000`, and the future by `4000`.

Note that sliding block decoders are well known for decoding binarydata. One well known technique for implementing sliding block decodersWith binary data is the Adler-Coppersmith-Hassner technique disclosed inU.S. Pat. No. 4,413,251. Another, related technique is disclosed in U.S.Pat. No. 4,882,583 to Dimitri et al. According to these techniques, thenumber of past symbols required in the window is fixed based on the codedesign. The number of future symbols should be maintained as small aspossible. These goals apply to the M-ary sliding block decoder as well.

The lookup table contains a plurality of entries. In one embodiment,each entry contains a possible sequence of N code symbols y', where N isthe size of sliding window 704. In alternative embodiments, each entryof the lookup table is addressed by (directly, indirectly or via adecoding scheme) one of the possible sequences of code symbols.

Based on the specific design of coder 304, there are a limited number ofpossible occurrences of N code symbols y'. The lookup table comprises asufficient number of entries such that there is one entry for each ofthe possible occurrences of N code symbols. As stated above, N is chosensuch that for each sequence of N code symbols y' an output bit set x' isunambiguously defined.

For the encoder 304 described above with reference to Table 1, thepreferred lookup table has 1252 entries and is illustrated in Table 3.For each entry of twelve code symbols y', there is an output bit set x'.Thus, to decode code symbols y', twelve consecutive code symbols y' areloaded into sliding window 704. The entry corresponding to those twelvecode symbols y' is located in the lookup table and the correspondingdata bit set x' is output. To determine the next output data bit set x',sliding window 704 is `slid` one code symbol set y into the future(i.e., in one embodiment, a next code symbol set y' is shifted intowindow 704 and the oldest shifted out) and the lookup process isrepeated. This sliding process continues as long as code symbols y' areprovided to sliding block decoder 608.

In one embodiment, the contents of window 704 are real-valued and,therefore, the table entry retrieved is the entry that is closest insquared Euclidean distance. Specifically, for a given window w=(w₁, . .. , w₁₂), the distance d_(j) for each table entry t_(j) is computed as:##EQU1## where, t_(ji) is the i'th component of the j'th table entry.The window w is then decoded to the table entry with the minimumdistance d_(j).

Other embodiments are contemplated where the contents of window 704 areused as an address or used to generate an address of an entry in amemory, where the contents of that entry contain the appropriate outputdata bit set x'.

The complexity of the decoding process can be reduced significantly bymaking hard decisions regarding read signal amplitudes. Specifically, inone embodiment, the read signal amplitudes z' are rounded-off orquantized to be one of the five permissible amplitude levels. Eventhough this results in a decrease in the performance of the decoder itcan significantly reduce its complexity: that is, the table can be atrue lookup table requiring no distance calculations.

It should be noted that at the beginning and the end of a data stream,there is a period of time during which window 704 is not full.

                                      TABLE 3                                     __________________________________________________________________________    Sequence of Code Symbols                                                                  x' Sequence of Code Symbols                                                                  x' Sequence of Code Symbols                                                                  x'                                  __________________________________________________________________________    000001000100                                                                              000                                                                              000020001000                                                                              100                                                                              000200010002                                                                              011                                 000001000300                                                                              000                                                                              000020003000                                                                              100                                                                              000200010003                                                                              011                                 000001000000                                                                              001                                                                              000040002000                                                                              101                                                                              000200010004                                                                              011                                 000001000001                                                                              001                                                                              000040004000                                                                              101                                                                              000200001000                                                                              100                                 000001000002                                                                              001                                                                              000040000200                                                                              110                                                                              000200003000                                                                              100                                 000001000003                                                                              001                                                                              000040000400                                                                              110                                                                              000200020000                                                                              101                                 000001000004                                                                              001                                                                              000040000100                                                                              111                                                                              000200020001                                                                              101                                 000001000010                                                                              010                                                                              000040000300                                                                              111                                                                              000200020002                                                                              101                                 000001000030                                                                              010                                                                              000100000100                                                                              000                                                                              000200020003                                                                              101                                 000001000020                                                                              010                                                                              000100000300                                                                              000                                                                              000200020004                                                                              101                                 000001000040                                                                              010                                                                              000100000200                                                                              001                                                                              000200030000                                                                              110                                 000003000100                                                                              011                                                                              000100000400                                                                              001                                     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                                                      030004000001                                                                              101                                                                              020000040003                                                                              111                                 030000030001                                                                              110                                                                              030004000002                                                                              101                                                                              020000040004                                                                              111                                 030000030002                                                                              110                                                                              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                         110                                                                              020000100002                                                                              000                                 030000040000                                                                              111                                                                              030004000030                                                                              110                                                                              020000100003                                                                              000                                 030000040001                                                                              111                                                                              030004000020                                                                              110                                     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 011                                 020000300000                                                                              010                                                                              020002000020                                                                              010                                                                              040000300040                                                                              011                                 020000300001                                                                              010                                                                              020002000040                                                                              010                                                                              040000200000                                                                              100                                 020000300002                                                                              010                                                                              020004000100                                                                              011                                                                              040000200001                                                                              100                                 020000300003                                                                              010                                                                              020004000300                                                                              011                                                                              040000200002                                                                              100                                 020000300004                                                                              010                                                                              020002000200                                                                              100                                                                              040000200003                                                                              100                                 020000300010                                                                              011                                                                              020002000400                                                                              100                                                                              040000200004                                                                              100                                 020000300030                                                                              011                                                              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                                         101                                                                              040000200020                                                                              101                                 020000200000                                                                              100                                                                              020004000003                                                                              101                                                                              040000200040                                                                              101                                 020000200001                                                                              100                                                                              020004000004                                                                              101                     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040000400002                                                                              110                                 020000200004                                                                              100                                                                              020004000020                                                                              110                                                                              040000400003                                                                              110                                 020000200010                                                                              101                                                                              020004000040                                                                              110                                                                              040000400004                                                                              110                                 020000200030                                                                              101                                                                              020004000200                                                                              111                                                                              040000400010                                                                              111                                 020000200020                                                                              101                                                                              020004000400                                                                              111                                                                              040000400030                                                                              111                                 020000200040                                                                              101                                                                              040000000100                                                                              000                                                                              040000400020                                                                              111                                 020000400000                                                                              110                                                                              040000000300                                                                              000                                                                              040000400040                                                                              111                                 020000400001                                                                              110                                                                              040000000200                                                                              001                                                                              040002000100                                                                              000                                 020000400002                                                                              110                                                                              040000000400                                                                              001                                                                              040002000300                                                                              000                                 020000400003                                                                              110                                     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 011                                                                              040002000004                                                                              001                                 020000400040                                                                              111                                                                              040000010003                                                                              011                                                                              040002000010                                                                              010                                 040001000100                                                                              000                                                                              040000010004                                                                              011                                                             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                                                      040000030001                                                                              110                                                                              040004000002                                                                              101                                 040001000040                                                                              010                                                                              040000030002                                                                              110                                                                              040004000003                                                                              101                                 040003000100                                                                              011                                                                              040000030003                                                                              110                                                                              040004000004                                                                              101                                 040003000300                                                                              011                                                                              040000030004                                                                              110                                                                              040004000010                                                                              110                                 040001000200                                                                              100                                                                              040000040000                                                                              111                                                                              040004000030                                                                              110                                 040001000400                                                                              100                                                                              040000040001                                                                              111                                                                              040004000020                                                                              110                                 040003000000                                                                              101                                                                              040000040002                                                                              111                                     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 000                                 040003000010                                                                              110                                                                              040000100002                                                                              000                                                                              200001000000                                                                              001                                 040003000030                                                                              110                                                                              040000100003                                                                              000                                                                              200001000001                                                                              001                                 040003000020                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111                                                                              040000100030                                                                              001                                                                              200001000004                                                                              001                                 040003000400                                                                              111                                                                              040000100020                                                                              001                                                                              200001000010                                                                              010                                 020002000100                                                                              000                                                              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400004000020                                                                              110                                 200003000000                                                                              101                                                                              400010003000                                                                              100                                                                              400004000040                                                                              110                                 200003000001                                                                              101                                                                              400030002000                                                                              101                                                                              400004000200                                                                              111                                 200003000002                                                                              101                                                                              400030004000                                                                              101                                                                              400004000400                                                                              111                                 200003000003                                                                              101                                                                              400030000200                                                                              110                                                                              400001000100                                                                              000                                 200003000004                                                                              101                                                                              400030000400                                                                              110                                                                              400001000300                                                                              000                                 200003000010                                                                              110                                                                              400030000100                                                                              111                                                                              400001000000                                                                              001                                 200003000030                                                                              110                                                                              400030000300                                                                              111                                                                              400001000000                                                                              001                                 200003000020                                                                              110                                                                              200010000100                                                                              000                                                                              400001000002                                                                              001                                 200003000040                                                                              110                                     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                                          100                                 200002000004                                                                              001                                                                              200030002000                                                                              101                                                                              400001000400                                                                              100                                 200002000010                                                                              010                                                                              200030004000                                                                              101                                                                              400003000000                                                                              101                    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                                      010                                                                              200030000100                                                                              111                                                                              400003000003                                                                              101                                 200004000100                                                                              011                                                                              200030000300                                                                              111                                                                              400003000004                                                                              101                                 200004000300                                                                              011                                                                              400020000100                                                                              000                                                                              400003000010                                                                              110                                 200002000200                                                                              100                                                                              400020000300                                                                              000                                                                              400003000030                                                                              110                                 200002000400                                                                              100                                                                              400020000200                                                                              001                                                                              400003000020                                                                              110                                 200004000000                                                                              101                                                                              400020000400                                                                              001                                                                              400003000040                                                                              110                                 200004000001                                                                              101                                                                              400020002000                                                                              010                                                                              400003000200                                                                              111                                 200004000002                                                                              101                                                                              400020004000                                                                              010                                                                              400003000400                                                                              111                                 200004000003                                                                              101                                                                              400040001000                                                                              011                                     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                                                                  001                                 200004000030                                                                              110                                                                              400020003000                                                                              100                                                                              100001000001                                                                              003\1                               200004000020                                                                              110                                                                              400040002000                                                                              101                                                                              100001000002                                                                              001                                 200004000040                                                                              110                                                                              400040004000                                                                              101                                                                              100001000003                                                                              001                                 200004000200                                                                              111                                                                              400040000200                                                                              110                                                                              100001000004                                                                              001                                 200004000400                                                                              111                                                                              400040000400                                                                              110                                                                              100001000010                                                                              010                                 200020000100                                                                              000                                                                              400040000100                                                                              111                                                                              100001000030                                                                              010                                 200020000300                                                                              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                400002000300                                                                              000                                                                              100003000100                                                                              011                                 200020002000                                                                              010                                                                              400002000000                                                                              001                                                                              100003000300                                                                              011                                 200020004000                                                                              010                                                                              400002000001                                                                              001                                                                              100001000200                                                                              100                                 200040001000                                                                              011                                                                              400002000002                                                                              001                                                                              100001000400                                                                              100                                 200040003000                                                                              011                                                                              400002000003                                                                              001                                                                              100003000000                                                                              101                                 200020001000                                                                              100                                                                              400002000004                                                                              001                                                                              100003000001                                                                              101                                 200020003000                                                                              100                                                                              400002000010                                                                              010                                                                              100003000002                                                                              101                                 200040002000                                                                              101                                                                              400002000030                                                                              010                                                                              100003000003                                                                              101                                 200040004000                                                                              101                                                                              400002000020                                                                              010                                                                              100003000004                                                     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200040000100                                                                              111                                                                              400004000300                                                                              011                                                                              100003000020                                                                              110                                 200040000300                                                                              111                                                                              400002000200                                                                              100                                                                              100003000040                                                                              110                                 400010000100                                                                              000                                                                              400002000400                                                                              100                                                                              100003000200                                                                              111                                 400010000300                                                                              000                                                                              400004000000                                                                              101                                                                              100003000400                                                                              111                                 400010000200                                                                              001                                                                              400004000001                                                                              101                                                                              100002000100                                                                              000                                 400010000400                                                                              001                                                                              400004000002                                                                              101                                                                              100002000300                                                                              000                                 400010002000                                                                              010                                                                              400004000003                                                                              101                                                                              100002000000                                                                              001                                 400010004000                                                                              010                                                                              400004000004                                                                              101                                                                              100002000001                                                                              001                                 400030001000                                                                              011                                                                              400004000010                                                                              110                                                                              100002000002                                                                              001                                 400030003000                                                                              011                                                                              400004000030                                                                              110                                                                              100002000003                                                                              001                                 100002000004                                                                              001                                                                              300030002000                                                                              101                                                             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                                          010                                 100002000020                                                                              010                                                                              300030000400                                                                              110                                                                              300004000100                                                                              011                                 100002000040                                                                              010                                                                              300030000100                                                                              111                                                                              300004000300                                                                              011                    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                                      100                                                                              100010000300                                                                              000                                                                              300004000000                                                                              101                                 100002000400                                                                              100                                                                              100010000200                                                                              001                                                                              300004000001                                                                              101                                 100004000000                                                                              101                                                                              100010000400                                                                              001                                                                              300004000002                                                                              101                                 100004000001                                                                              101                                                                              100010002000                                                                              010                                                                              300004000003                                                                              101                                 100004000002                                                                              101                                                                              100010004000                                                                              010                                                                              300004000004                                                                              101                                 100004000003                                                                              101                                                                              100030001000                                                                              011                                                                              300004000010                                                                              110                                 100004000004                                                                              101                                                                              100030003000                                                                              011                                                                              300004000030                                                                              110                                 100004000010                                                                              110                                                                              100010001000                                                                              100                                                                              300004000020                                                                              110                                 100004000030                                                                              110                                                                              100010003000                                                                              100                                                                              300004000040                                                                              110                                 100004000020                                                                              110                                                                              100030002000                                                                              101                                                                              300004000200                                                                              111                                 100004000040                                                                              110                                                                              100030004000                                                                              101                                                                              300004000400                                                                              111                                 100004000200                                                                              111                                                                              100030000200                                                                              110                                                                              300001000100                                                                              000                                 100004000400                                                                              111                                                                              100030000400                                                                              110                                                                              300001000300                                                                              000                                 100020000100                                                                              000                                                                              100030000100                                                                              111                                                                              300001000000                                                                              001                                 100020000300                                                                              000                                                                              100030000300                                                                              111                                                                              300001000001                                                                              001                                 100020000200                                                                              001                                                                              300020000100                                                                              000                                                                              300001000002                                                                              001                                 100020000400                                                                              001                                                                              300020000300                                                                              000                                                                              300001000003                                                                              001                                 100020002000                                                                              010                                                                              300020000200                                                                              001                                                                              300001000004                                                                              001                                 100020004000                                                                              010                                                                              300020000400                                                                              001                                                                              300001000010                                                                              010                                 100040001000                                                                              011                                                                              300020002000                                                                              010                                                                              300001000030                                                                              010                                 100040003000                                                                              011                                                                              300020004000                                                                              010                                                                              300001000020                                                                              010                                 100020001000                                                                              100                                                                              300040001000                                                                              011                                                                              300001000040                                                                              010                                 100020003000                                                                              100                                                                              300040003000                                                                              011                                                                              300003000100                                                                              011                                 100040002000                                                                              101                                                                              300020001000                                                                              100                                                                              300003000300                                                                              011                                 100040004000                                                                              101                                                                              300020003000                                                                              100                                                                              300001000200                                                                              100                                 100040000200                                                                              110                                                                              300040002000                                                                              101                                                                              300001000400                                                                              100                                 100040000400                                                                              110                                                                              300040004000                                                                              101                                                                              300003000000                                                                              101                                 100040000100                                                                              111                                                                              300040000200                                                                              110                                                                              300003000001                                                                              101                                 100040000300                                                                              111                                                                              300040000400                                                                              110                                                                              300003000002                                                                              101                                 300010000100                                                                              000                                                                              300040000100                                                                              111                                                                              300003000003                                                                              101                                 300010000300                                                                              000                                                                              300040000300                                                                              111                                                                              300003000004                                                                              101                                 300010000200                                                                              001                                                                              300002000100                                                                              000                                                                              300003000010                                                                              110                                 300010000400                                                                              001                                                                              300002000300                                                                              000                                                                              300003000030                                                                              110                                 300010002000                                                                              010                                                                              300002000000                                                                              001                                                                              300003000020                                                                              110                                 300010004000                                                                              010                                                                              300002000001                                                                              001                                                                              300003000040                                                                              110                                 300030001000                                                                              011                                                                              300002000002                                                                              001                                                                              300003000200                                                                              111                                 300030003000                                                                              011                                                                              300002000003                                                                              001                                                                              300003000400                                                                              111                                 300010001000                                                                              100                                                                              300002000004                                                                              001                                                300010003000                                                                              100                                                                              300002000010                                                                              010                                                __________________________________________________________________________

7. Conclusion

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. Thus, the breadth and scope of thepresent invention should not be limited by any of the above-describedexemplary embodiments, but should be defined only in accordance with thefollowing claims and their equivalents.

What is claimed is:
 1. A method for encoding a series of binary input data bits x into an M=5 run-length limited (3,7) code having a rate R=3/4, to produce a series of code symbols y, the method comprising the steps of receiving at a six-state encoder an input data bit set x_(i) . . . x_(i+2) to be encoded into the M=5 run-length limited (3,7) code, and generating in said encoder a code symbol set y_(i) . . . y_(i+3) based on said input data bit set x_(i) . . . x_(i+2) according to the following mapping table:

    ______________________________________                                         INPUT         OUTPUT   NEXT STATE                                              ______________________________________                                         STATE 0                                                                        000           0000     2                                                       001           0000     3                                                       010           0000     4                                                       011           0001     0                                                       100           0000     5                                                       101           0002     0                                                       110           0003     0                                                       111           0004     0                                                       STATE 1                                                                        000           0010     0                                                       001           0010     1                                                       010           0030     0                                                       011           0030     1                                                       100           0020     0                                                       101           0020     1                                                       110           0040     0                                                       111           0040     1                                                       STATE 2                                                                        000           0100     2                                                       001           0100     0                                                       010           0100     1                                                       011           0300     2                                                       100           0100     3                                                       101           0300     0                                                       110           0300     1                                                       111           0300     3                                                       STATE 3                                                                        000           0200     2                                                       001           0200     0                                                       010           0200     1                                                       011           0400     2                                                       100           0200     3                                                       101           0400     0                                                       110           0400     1                                                       111           0400     3                                                       STATE 4                                                                        000           2000     2                                                       001           2000     3                                                       010           2000     4                                                       011           4000     5                                                       100           2000     5                                                       101           4000     4                                                       110           4000     3                                                       111           4000     2                                                       STATE 5                                                                        000           1000     2                                                       001           1000     3                                                       010           1000     4                                                       011           3000     5                                                       100           1000     5                                                       101           3000     4                                                       110           3000     3                                                       111           3000     2                                                       ______________________________________                                    

wherein said code symbol set y_(i) . . . y_(i+3) is generated based on the input data bit set x_(i) . . . x_(i+2) and the current state of said six-state encoder when said data bit set x_(i) . . . x_(i+2) is received.
 2. A method for encoding a series of binary input data bits x into an M-ary run-length limited (3,7) code having a rate R=3/4 to produce a series of code symbols y, each code symbol being at one of M levels, where M>2 and each pair of non-zero code symbols having a minimum of three and a maximum of seven zeroes therebetween, the method comprising the steps of receiving at a six-state encoder an input data bit set x_(i) . . . x_(i+2) to be encoded into the M-ary run-length limited (3,7) code, and generating in said encoder a code symbol set y_(i) . . . y_(i+3) based on said input data bit set x_(i) . . . x_(i+2) according to the following mapping table:

    ______________________________________                                         INPUT         OUTPUT   NEXT STATE                                              ______________________________________                                         STATE 0                                                                        000           0000     2                                                       001           0000     3                                                       010           0000     4                                                       011           0001     0                                                       100           0000     5                                                       101           0002     0                                                       110           0003     0                                                       111           0004     0                                                       STATE 1                                                                        000           0010     0                                                       001           0010     1                                                       010           0030     0                                                       011           0030     1                                                       100           0020     0                                                       101           0020     1                                                       110           0040     0                                                       111           0040     1                                                       STATE 2                                                                        000           0100     2                                                       001           0100     0                                                       010           0100     1                                                       011           0300     2                                                       100           0100     3                                                       101           0300     0                                                       110           0300     1                                                       111           0300     3                                                       STATE 3                                                                        000           0200     2                                                       001           0200     0                                                       010           0200     1                                                       011           0400     2                                                       100           0200     3                                                       101           0400     0                                                       110           0400     1                                                       111           0400     3                                                       STATE 4                                                                        000           2000     2                                                       001           2000     3                                                       010           2000     4                                                       011           4000     5                                                       100           2000     5                                                       101           4000     4                                                       110           4000     3                                                       111           4000     2                                                       STATE 5                                                                        000           1000     2                                                       001           1000     3                                                       010           1000     4                                                       011           3000     5                                                       100           1000     5                                                       101           3000     4                                                       110           3000     3                                                       111           3000     2                                                       ______________________________________                                    

wherein said code symbol set y_(i) . . . y_(i+3) is generated based on the input data bit set x_(i) . . . x_(i+2) and the current state of said six-state encoder when said data bit set x_(i) . . . x_(i+2) is received.
 3. A method for converting binary input data bits into a series of waveform amplitudes each amplitude being at one of M=5 levels, the method comprising the steps of:encoding the binary input data bits into an M=5 (3,7) code having a rate R=3/4 to produce a series of code symbols, each code symbol being at one of M levels, and each pair of non-zero code symbols having a minimum of three and a maximum of seven zeroes therebetween, said encoding step comprising the step of generating a code symbol set y_(i) . . . y_(i+3) for each input data bit set x_(i) . . . x_(i+2) according to the following mapping table:

    ______________________________________                                         INPUT         OUTPUT   NEXT STATE                                              ______________________________________                                         STATE 0                                                                        000           0000     2                                                       001           0000     3                                                       010           0000     4                                                       011           0001     0                                                       100           0000     5                                                       101           0002     0                                                       110           0003     0                                                       111           0004     0                                                       STATE 1                                                                        000           0010     0                                                       001           0010     1                                                       010           0030     0                                                       011           0030     1                                                       100           0020     0                                                       101           0020     1                                                       110           0040     0                                                       111           0040     1                                                       STATE 2                                                                        000           0100     2                                                       001           0100     0                                                       010           0100     1                                                       011           0300     2                                                       100           0100     3                                                       101           0300     0                                                       110           0300     1                                                       111           0300     3                                                       STATE 3                                                                        000           0200     2                                                       001           0200     0                                                       010           0200     1                                                       011           0400     2                                                       100           0200     3                                                       101           0400     0                                                       110           0400     1                                                       111           0400     3                                                       STATE 4                                                                        000           2000     2                                                       001           2000     3                                                       010           2000     4                                                       011           4000     5                                                       100           2000     5                                                       101           4000     4                                                       110           4000     3                                                       111           4000     2                                                       STATE 5                                                                        000           1000     2                                                       001           1000     3                                                       010           1000     4                                                       011           3000     5                                                       100           1000     5                                                       101           3000     4                                                       110           3000     3                                                       111           3000     2                                                       ______________________________________                                    

wherein said code symbol set y_(i) . . . y_(i+3) is generated based on the input data bit set x_(i) . . . x_(i+2) and the current state of said six-state encoder when said data bit set x_(i) . . . x_(i+2) is received; and generating in a waveform encoder waveform signal amplitudes based on said code symbols, wherein each said waveform signal amplitude is generated by modulo five addition of a current code symbol with a previous waveform signal amplitude.
 4. In a data storage system having storage media capable of storing data at multiple levels, a method for converting binary input data bits into a series of waveform amplitudes each amplitude being at one of M=5 levels and storing the waveform amplitudes on the storage medium, the method comprising the steps of:encoding the binary input data bits into an M=5 (3,7) code having a rate R=3/4 to produce a series of code symbols, each code symbol being at one of M levels, and each pair of non-zero code symbols having a minimum of three and a maximum of seven zeroes therebetween, comprising the step of generating a code symbol set y_(i) . . . y_(i+3) for each input data bit set x_(i) . . . x_(i+2) according to the following mapping table:

    ______________________________________                                         INPUT         OUTPUT   NEXT STATE                                              ______________________________________                                         STATE 0                                                                        000           0000     2                                                       001           0000     3                                                       010           0000     4                                                       011           0001     0                                                       100           0000     5                                                       101           0002     0                                                       110           0003     0                                                       111           0004     0                                                       STATE 1                                                                        000           0010     0                                                       001           0010     1                                                       010           0030     0                                                       011           0030     1                                                       100           0020     0                                                       101           0020     1                                                       110           0040     0                                                       111           0040     1                                                       STATE 2                                                                        000           0100     2                                                       001           0100     0                                                       010           0100     1                                                       011           0300     2                                                       100           0100     3                                                       101           0300     0                                                       110           0300     1                                                       111           0300     3                                                       STATE 3                                                                        000           0200     2                                                       001           0200     0                                                       010           0200     1                                                       011           0400     2                                                       100           0200     3                                                       101           0400     0                                                       110           0400     1                                                       111           0400     3                                                       STATE 4                                                                        000           2000     2                                                       001           2000     3                                                       010           2000     4                                                       011           4000     5                                                       100           2000     5                                                       101           4000     4                                                       110           4000     3                                                       111           4000     2                                                       STATE 5                                                                        000           1000     2                                                       001           1000     3                                                       010           1000     4                                                       011           3000     5                                                       100           1000     5                                                       101           3000     4                                                       110           3000     3                                                       111           3000     2                                                       ______________________________________                                    

wherein said code symbol set y_(i) . . . y_(i+3) is generated based on the input data bit set x_(i) . . . x_(i+2) and the current state of said six-state encoder when said data bit set x_(i) . . . x_(i+2) is received; generating in a waveform encoder waveform signal amplitudes based on said code symbols, wherein each said waveform signal amplitude is generated by modulo five addition of a current code symbol with a previous waveform signal amplitude; and modulating a write signal with said waveform signal amplitudes to write a series of features to the storage medium, each feature being at a level corresponding to a waveform signal amplitude.
 5. The method according to claim 4, wherein said waveform signal amplitudes modulate a write laser and wherein said modulated write laser writes features onto the storage medium.
 6. The method according to claim 4, further comprising the steps of:reading the storage medium to recover output waveform signal amplitudes; converting said output waveform signal amplitudes into output code symbols, wherein each output code symbol is at one of five levels; and decoding said output code symbols to obtain output data bits, wherein said output data bits correspond to the digital input data.
 7. The method of claim 6, wherein said step of decoding comprises the steps of:identifying with a sliding window a sequence of code symbols that represents a given code symbol set; and in a table comprising a number of entries corresponding to a number of possible unique sequences of code symbols, each entry identified by one of said possible unique sequences of code symbols, and wherein each entry identifies an output data bit set that is a decoded data bit set for the code symbol set represented by the sequence of code symbols that identifies that entry, looking up with said identified sequence of code symbols the output data bit for said given code symbol.
 8. The method according to claim 7, wherein said lookup table provides an output bit set x' for each sequence of code symbols y' according to the following table:

    __________________________________________________________________________     Sequence of Code Symbols                                                                   x' Sequence of Code Symbols                                                                   x' Sequence of Code Symbol                                                                    x'                                   __________________________________________________________________________     000001000100                                                                               000                                                                               000020001000                                                                               100                                                                               000200010002                                                                               011                                  000001000300                                                                               000                                                                               000020003000                                                                               100                                                                               000200010003                                                                               011                                  000001000000                                                                               001                                                                               000040002000                                                                               101                                                                               000200010004                                                                               011                                  000001000001                                                                               001                                                                               000040004000                                                                               101                                                                               000200001000                                                                               100                                  000001000002                                                                               001                                                                               000040000200                                                                               110                                                                               000200003000                                                                               100                                  000001000003                                                                               001                                                                               000040000400                                                                               110                                                                               000200020000                                                                               101                                  000001000004                                                                               001                                                                               000040000100                                                                               111                                                                               000200020001                                                                               101                                  000001000010                                                                               010                                                                               000040000300                                                                               111                                                                               000200020002                                                                               101                                  000001000030                                                                               010                                                                               000100000100                                                                               000                                                                               000200020003                                                                               101                                  000001000020                                                                               010                                                                               000100000300                                                                               000                                                                               000200020004                                                                               101                                  000001000040                                                                               010                                                                               000100000200                                                                               001                                                                               000200030000                                                                               110                                  000003000100                                                                               011                                                                               000100000400                                                                               001                                                                               000200030001                                                                               110                                  000003000300                                                                               011                                                                               000100002000                                                                               010                                                                               000200030002                                                                               110                                  000001000200                                                                               100                                                                               000100004000                                                                               010                                                                               000200030003                                                                               110                                  000001000400                                                                               100                                                                               000100010000                                                                               011                                                                               000200030004                                                                               110                                  000003000000                                                                               101                                                                               000100010001                                                                               011                                                                               000200040000                                                                               111                                  000003000001                                                                               101                                                                               000100010002                                                                               011                                                                               000200040001                                                                               111                                  000003000002                                                                               101                                                                               000100010003                                                                               011                                                                               000200040002                                                                               111                                  000003000003                                                                               101                                                                               000100010004                                                                               011                                                                               000200040003                                                                               111                                  000003000004                                                                               101                                                                               000100001000                                                                               100                                                                               000200040004                                                                               111                                  000003000010                                                                               110                                                                               000100003000                                                                               100                                                                               000300000100                                                                               000                                  000003000030                                                                               110                                                                               000100020000                                                                               101                                                                               000300000300                                                                               000                                  000003000020                                                                               110                                                                               000100020001                                                                               101                                                                               000300000200                                                                               001                                  000003000040                                                                               110                                                                               000100020002                                                                               101                                                                               000300000400                                                                               001                                  000003000200                                                                               111                                                                               000100020003                                                                               101                                                                               000300002000                                                                               010                                  000003000400                                                                               111                                                                               000100020004                                                                               101                                                                               000300004000                                                                               010                                  000002000100                                                                               000                                                                               000100030000                                                                               110                                                                               000300010000                                                                               011                                  000002000300                                                                               000                                                                               000100030001                                                                               110                                                                               000300010001                                                                               011                                  000002000000                                                                               001                                                                               000100030002                                                                               110                                                                               000300010002                                                                               011                                  000002000001                                                                               001                                                                               000100030003                                                                               110                                                                               000300010003                                                                               011                                  000002000002                                                                               001                                                                               000100030004                                                                               110                                                                               000300010004                                                                               011                                  000002000003                                                                               001                                                                               000100040000                                                                               111                                                                               000300001000                                                                               100                                  000002000004                                                                               001                                                                               000100040001                                                                               111                                                                               000300003000                                                                               100                                  000002000010                                                                               010                                                                               000100040002                                                                               111                                                                               000300020000                                                                               101                                  000002000030                                                                               010                                                                               000100040003                                                                               111                                                                               000300020001                                                                               101                                  000002000020                                                                               010                                                                               000100040004                                                                               111                                                                               000300020002                                                                               101                                  000002000040                                                                               010                                                                               000010000100                                                                               000                                                                               000300020003                                                                               101                                  000004000100                                                                               011                                                                               000010000300                                                                               000                                                                               000300020004                                                                               101                                  000004000300                                                                               011                                                                               000010000200                                                                               001                                                                               000300030000                                                                               110                                  000002000200                                                                               100                                                                               000010000400                                                                               001                                                                               000300030001                                                                               110                                  000002000400                                                                               100                                                                               000010002000                                                                               010                                                                               000300030002                                                                               110                                  000004000000                                                                               101                                                                               000010004000                                                                               010                                                                               000300030003                                                                               110                                  000004000001                                                                               101                                                                               000030001000                                                                               011                                                                               000300030004                                                                               110                                  000004000002                                                                               101                                                                               000030003000                                                                               011                                                                               000300040000                                                                               111                                  000004000003                                                                               101                                                                               000010001000                                                                               100                                                                               000300040001                                                                               111                                  000004000004                                                                               101                                                                               000010003000                                                                               100                                                                               000300040002                                                                               111                                  000004000010                                                                               110                                                                               000030002000                                                                               101                                                                               000300040003                                                                               111                                  000004000030                                                                               110                                                                               000030004000                                                                               101                                                                               000300040004                                                                               111                                  000004000020                                                                               110                                                                               000030000200                                                                               110                                                                               000400000100                                                                               000                                  000004000040                                                                               110                                                                               000030000400                                                                               110                                                                               000400000300                                                                               000                                  000004000200                                                                               111                                                                               000030000100                                                                               111                                                                               000400000200                                                                               001                                  000004000400                                                                               111                                                                               000030000300                                                                               111                                                                               000400000400                                                                               001                                  000020000100                                                                               000                                                                               000200000100                                                                               000                                                                               000400002000                                                                               010                                  000020000300                                                                               000                                                                               000200000300                                                                               000                                                                               000400004000                                                                               010                                  000020000200                                                                               001                                                                               000200000200                                                                               001                                                                               000400010000                                                                               011                                  000020000400                                                                               001                                                                               000200000400                                                                               001                                                                               000400010001                                                                               011                                  000020002000                                                                               010                                                                               000200002000                                                                               010                                                                               000400010002                                                                               011                                  000020004000                                                                               010                                                                               000200004000                                                                               010                                                                               000400010003                                                                               011                                  000040001000                                                                               011                                                                               000200010000                                                                               011                                                                               000400010004                                                                               011                                  000040003000                                                                               011                                                                               000200010001                                                                               011                                                                               000400001000                                                                               100                                  000400003000                                                                               100                                                                               001000200002                                                                               100                                                                               003000200002                                                                               100                                  000400020000                                                                               101                                                                               001000200003                                                                               100                                                                               003000200003                                                                               100                                  000400020001                                                                               101                                                                               001000200004                                                                               100                                                                               003000200004                                                                               100                                  000400020002                                                                               101                                                                               001000200010                                                                               101                                                                               003000200010                                                                               101                                  000400020003                                                                               101                                                                               001000200030                                                                               101                                                                               003000200030                                                                               101                                  000400020004                                                                               101                                                                               001000200020                                                                               101                                                                               003000200020                                                                               101                                  000400030000                                                                               110                                                                               001000200040                                                                               101                                                                               003000200040                                                                               101                                  000400030001                                                                               110                                                                               001000400000                                                                               110                                                                               003000400000                                                                               110                                  000400030002                                                                               110                                                                               001000400001                                                                               110                                                                               003000400001                                                                               110                                  000400030003                                                                               110                                                                               001000400002                                                                               110                                                                               003000400002                                                                               110                                  000400030004                                                                               110                                                                               001000400003                                                                               110                                                                               003000400003                                                                               110                                  000400040000                                                                               111                                                                               001000400004                                                                               110                                                                               003000400004                                                                               110                                  000400040001                                                                               111                                                                               001000400010                                                                               111                                                                               003000400010                                                                               111                                  000400040002                                                                               111                                                                               001000400030                                                                               111                                                                               003000400030                                                                               111                                  000400040003                                                                               111                                                                               001000400020                                                                               111                                                                               003000400020                                                                               111                                  000400040004                                                                               111                                                                               001000400040                                                                               111                                                                               003000400040                                                                               111                                  001000000100                                                                               000                                                                               003000000100                                                                               000                                                                               002000000100                                                                               000                                  001000000300                                                                               000                                                                               003000000300                                                                               000                                                                               002000000300                                                                               000                                  001000000200                                                                               001                                                                               003000000200                                                                               001                                                                               002000000200                                                                               001                                  001000000400                                                                               001                                                                               003000000400                                                                               001                                                                               002000000400                                                                               001                                  001000002000                                                                               010                                                                               003000002000                                                                               010                                                                               002000002000                                                                               010                                  001000004000                                                                               010                                                                               003000004000                                                                               010                                                                               002000004000                                                                               010                                  001000010000                                                                               011                                                                               003000010000                                                                               011                                                                               002000010000                                                                               011                                  001000010001                                                                               011                                                                               003000010001                                                                               011                                                                               002000010001                                                                               011                                  001000010002                                                                               011                                                                               003000010002                                                                               011                                                                               002000010002                                                                               011                                  001000010003                                                                               011                                                                               003000010003                                                                               011                                                                               002000010003                                                                               011                                  001000010004                                                                               011                                                                               003000010004                                                                               011                                                                               002000010004                                                                               011                                  001000001000                                                                               100                                                                               003000001000                                                                               100                                                                               002000001000                                                                               100                                  001000003000                                                                               100                                                                               003000003000                                                                               100                                                                               002000003000                                                                               100                                  001000020000                                                                               101                                                                               003000020000                                                                               101                                                                               002000020000                                                                               101                                  001000020001                                                                               101                                                                               003000020001                                                                               101                                                                               002000020001                                                                               101                                  001000020002                                                                               101                                                                               003000020002                                                                               101                                                                               002000020002                                                                               101                                  001000020003                                                                               101                                                                               003000020003                                                                               101                                                                               002000020003                                                                               101                                  001000020004                                                                               101                                                                               003000020004                                                                               101                                                                               002000020004                                                                               101                                  001000030000                                                                               110                                                                               003000030000                                                                               110                                                                               002000030000                                                                               110                                  001000030001                                                                               110                                                                               003000030001                                                                               110                                                                               002000030001                                                                               110                                  001000030002                                                                               110                                                                               003000030002                                                                               110                                                                               002000030002                                                                               110                                  001000030003                                                                               110                                                                               003000030003                                                                               110                                                                               002000030003                                                                               110                                  001000030004                                                                               110                                                                               003000030004                                                                               110                                                                               002000030004                                                                               110                                  001000040000                                                                               111                                                                               003000040000                                                                               111                                                                               002000040000                                                                               111                                  001000040001                                                                               111                                                                               003000040001                                                                               111                                                                               002000040001                                                                               111                                  001000040002                                                                               111                                                                               003000040002                                                                               111                                                                               002000040002                                                                               111                                  001000040003                                                                               111                                                                               003000040003                                                                               111                                                                               002000040003                                                                               111                                  001000040004                                                                               111                                                                               003000040004                                                                               111                                                                               002000040004                                                                               111                                  001000100000                                                                               000                                                                               003000100000                                                                               000                                                                               002000100000                                                                               000                                  001000100001                                                                               000                                                                               003000100001                                                                               000                                                                               002000100001                                                                               000                                  001000100002                                                                               000                                                                               003000100002                                                                               000                                                                               002000100002                                                                               000                                  001000100003                                                                               000                                                                               003000100003                                                                               000                                                                               002000100003                                                                               000                                  001000100004                                                                               000                                                                               003000100004                                                                               000                                                                               002000100004                                                                               000                                  001000100010                                                                               001                                                                               003000100010                                                                               001                                                                               002000100010                                                                               001                                  001000100030                                                                               001                                                                               003000100030                                                                               001                                                                               002000100030                                                                               001                                  001000100020                                                                               001                                                                               003000100020                                                                               001                                                                               002000100020                                                                               001                                  001000100040                                                                               001                                                                               003000100040                                                                               001                                                                               002000100040                                                                               001                                  001000300000                                                                               010                                                                               003000300000                                                                               010                                                                               002000300000                                                                               010                                  001000300001                                                                               010                                                                               003000300001                                                                               010                                                                               002000300001                                                                               010                                  001000300002                                                                               010                                                                               003000300002                                                                               010                                                                               002000300002                                                                               010                                  001000300003                                                                               010                                                                               003000300003                                                                               010                                                                               002000300003                                                                               010                                  001000300004                                                                               010                                                                               003000300004                                                                               010                                                                               002000300004                                                                               010                                  001000300010                                                                               011                                                                               003000300010                                                                               011                                                                               002000300010                                                                               011                                  001000300030                                                                               011                                                                               003000300030                                                                               011                                                                               002000300030                                                                               011                                  001000300020                                                                               011                                                                               003000300020                                                                               011                                                                               002000300020                                                                               011                                  001000300040                                                                               011                                                                               003000300040                                                                               011                                                                               002000300040                                                                               011                                  001000200000                                                                               100                                                                               003000200000                                                                               100                                                                               002000200000                                                                               100                                  001000200001                                                                               100                                                                               003000200001                                                                               100                                                                               002000200001                                                                               100                                  002000200002                                                                               100                                                                               004000200000                                                                               100                                                                               010000030000                                                                               110                                  002000200003                                                                               100                                                                               004000200001                                                                               100                                                                               010000030001                                                                               110                                  002000200004                                                                               100                                                                               004000200002                                                                               100                                                                               010000030002                                                                               110                                  002000200010                                                                               101                                                                               004000200003                                                                               100                                                                               010000030003                                                                               110                                  002000200030                                                                               101                                                                               004000200004                                                                               100                                                                               010000030004                                                                               110                                  002000200020                                                                               101                                                                               004000200010                                                                               101                                                                               010000040000                                                                               111                                  002000200040                                                                               101                                                                               004000200030                                                                               101                                                                               010000040001                                                                               111                                  002000400000                                                                               110                                                                               004000200020                                                                               101                                                                               010000040002                                                                               111                                  002000400001                                                                               110                                                                               004000200040                                                                               101                                                                               010000040003                                                                               111                                  002000400002                                                                               110                                                                               004000400000                                                                               110                                                                               010000040004                                                                               111                                  002000400003                                                                               110                                                                               004000400001                                                                               110                                                                               010000100000                                                                               000                                  002000400004                                                                               110                                                                               004000400002                                                                               110                                                                               010000100001                                                                               000                                  002000400010                                                                               111                                                                               004000400003                                                                               110                                                                               010000100002                                                                               000                                  002000400030                                                                               111                                                                               004000400004                                                                               110                                                                               010000100003                                                                               000                                  002000400020                                                                               111                                                                               004000400010                                                                               111                                                                               010000100004                                                                               000                                  002000400040                                                                               111                                                                               004000400030                                                                               111                                                                               010000100010                                                                               001                                  004000000100                                                                               000                                                                               004000400020                                                                               111                                                                               010000100030                                                                               001                                  004000000300                                                                               000                                                                               004000400040                                                                               111                                                                               010000100020                                                                               001                                  004000000200                                                                               001                                                                               010001000100                                                                               000                                                                               010000100040                                                                               001                                  004000000400                                                                               001                                                                               010001000300                                                                               000                                                                               010000300000                                                                               010                                  004000002000                                                                               010                                                                               010001000000                                                                               001                                                                               010000300001                                                                               010                                  004000004000                                                                               010                                                                               010001000001                                                                               001                                                                               010000300002                                                                               010                                  004000010000                                                                               011                                                                               010001000002                                                                               001                                                                               010000300003                                                                               010                                  004000010001                                                                               011                                                                               010001000003                                                                               001                                                                               010000300004                                                                               010                                  004000010002                                                                               011                                                                               010001000004                                                                               001                                                                               010000300010                                                                               011                                  004000010003                                                                               011                                                                               010001000010                                                                               010                                                                               010000300030                                                                               011                                  004000010004                                                                               011                                                                               010001000030                                                                               010                                                                               010000300020                                                                               011                                  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200040000200                                                                               110                                                                               400002000040                                                                               010                                                                               100003000010                                                                               110                                  200040000400                                                                               110                                                                               400004000100                                                                               011                                                                               100003000030                                                                               110                                  200040000100                                                                               111                                                                               400004000300                                                                               011                                                                               100003000020                                                                               110                                  200040000300                                                                               111                                                                               400002000200                                                                               100                                                                               100003000040                                                                               110                                  400010000100                                                                               000                                                                               400002000400                                                                               100                                                                               100003000200                                                                               111                                  400010000300                                                                               000                                                                               400004000000                                                                               101                                                                               100003000400                                                                               111                                  400010000200                                                                               001                                                                               400004000001                                                                               101                                                                               100002000100                                                                               000                                  400010000400                                                                               001                                                                               400004000002                                                                               101                                                                               100002000300                                                                               000                                  400010002000                                                                               010                                                                               400004000003                                                                               101                                                                               100002000000                                                                               001                                  400010004000                                                                               010                                                                               400004000004                                                                               101                                                                               100002000001                                                                               001                                  400030001000                                                                               011                                                                               400004000010                                                                               110                                                                               100002000002                                                                               001                                  400030003000                                                                               011                                                                               400004000030                                                                               110                                                                               100002000003                                                                               001                                  100002000004                                                                               001                                                                               300030002000                                                                               101                                                                               300002000030                                                                               010                                  100002000010                                                                               010                                                                               300030004000                                                                               101                                                                               300002000020                                                                               010                                  100002000030                                                                               010                                                                               300030000200                                                                               110                                                                               300002000040                                                                               010                                  100002000020                                                                               010                                                                               300030000400                                                                               110                                                                               300004000100                                                                               011                                  100002000040                                                                               010                                                                               300030000100                                                                               111                                                                               300004000300                                                                               011                                  100004000100                                                                               011                                                                               300030000300                                                                               111                                                                               300002000200                                                                               100                                  100004000300                                                                               011                                                                               100010000100                                                                               000                                                                               300002000400                                                                               100                                  100002000200                                                                               100                                                                               100010000300                                                                               000                                                                               300004000000                                                                               101                                  100002000400                                                                               100                                                                               100010000200                                                                               001                                                                               300004000001                                                                               101                                  100004000000                                                                               101                                                                               100010000400                                                                               001                                                                               300004000002                                                                               101                                  100004000001                                                                               101                                                                               100010002000                                                                               010                                                                               300004000003                                                                               101                                  100004000002                                                                               101                                                                               100010004000                                                                               010                                                                               300004000004                                                                               101                                  100004000003                                                                               101                                                                               100030001000                                                                               011                                                                               300004000010                                                                               110                                  100004000004                                                                               101                                                                               100030003000                                                                               011                                                                               300004000030                                                                               110                                  100004000010                                                                               110                                                                               100010001000                                                                               100                                                                               300004000020                                                                               110                                  100004000030                                                                               110                                                                               100010003000                                                                               100                                                                               300004000040                                                                               110                                  100004000020                                                                               110                                                                               100030002000                                                                               101                                                                               300004000200                                                                               111                                  100004000040                                                                               110                                                                               100030004000                                                                               101                                                                               300004000400                                                                               111                                  100004000200                                                                               111                                                                               100030000200                                                                               110                                                                               300001000100                                                                               000                                  100004000400                                                                               111                                                                               100030000400                                                                               110                                                                               300001000300                                                                               000                                  100020000100                                                                               000                                                                               100030000100                                                                               111                                                                               300001000000                                                                               001                                  100020000300                                                                               000                                                                               100030000300                                                                               111                                                                               300001000001                                                                               001                                  100020000200                                                                               001                                                                               300020000100                                                                               000                                                                               300001000002                                                                               001                                  100020000400                                                                               001                                                                               300020000300                                                                               000                                                                               300001000003                                                                               001                                  100020002000                                                                               010                                                                               300020000200                                                                               001                                                                               300001000004                                                                               001                                  100020004000                                                                               010                                                                               300020000400                                                                               001                                                                               300001000010                                                                               010                                  100040001000                                                                               011                                                                               300020002000                                                                               010                                                                               300001000030                                                                               010                                  100040003000                                                                               011                                                                               300020004000                                                                               010                                                                               300001000020                                                                               010                                  100020001000                                                                               100                                                                               300040001000                                                                               011                                                                               300001000040                                                                               010                                  100020003000                                                                               100                                                                               300040003000                                                                               011                                                                               300003000100                                                                               011                                  100040002000                                                                               101                                                                               300020001000                                                                               100                                                                               300003000300                                                                               011                                  100040004000                                                                               101                                                                               300020003000                                                                               100                                                                               300001000200                                                                               100                                  100040000200                                                                               110                                                                               300040002000                                                                               101                                                                               300001000400                                                                               100                                  100040000400                                                                               110                                                                               300040004000                                                                               101                                                                               300003000000                                                                               101                                  100040000100                                                                               111                                                                               300040000200                                                                               110                                                                               300003000001                                                                               101                                  100040000300                                                                               111                                                                               300040000400                                                                               110                                                                               300003000002                                                                               101                                  300010000100                                                                               000                                                                               300040000100                                                                               111                                                                               300003000003                                                                               101                                  300010000300                                                                               000                                                                               300040000300                                                                               111                                                                               300003000004                                                                               101                                  300010000200                                                                               001                                                                               300002000100                                                                               000                                                                               300003000010                                                                               110                                  300010000400                                                                               001                                                                               300002000300                                                                               000                                                                               300003000030                                                                               110                                  300010002000                                                                               010                                                                               300002000000                                                                               001                                                                               300003000020                                                                               110                                  300010004000                                                                               010                                                                               300002000001                                                                               001                                                                               300003000040                                                                               110                                  300030001000                                                                               011                                                                               300002000002                                                                               001                                                                               300003000200                                                                               111                                  300030003000                                                                               011                                                                               300002000003                                                                               001                                                                               300003000400                                                                               111                                  300010001000                                                                               100                                                                               300002000004                                                                               001                                                 300010003000                                                                               100                                                                               300002000010                                                                               010                                                 __________________________________________________________________________


9. The method of claim 7, wherein said look-up step comprises the step of determining, using a squared euclidean distance, which table entry is identified by a sequence of code symbols.
 10. A method for decoding data stored on a storage medium, wherein said stored data has been encoding using an M=5 (3,7) code, comprising the steps of:reading the storage medium to recover output waveform signal amplitudes; converting said output waveform signal amplitudes into output code symbols, wherein each output code symbol is at one of five levels; and decoding said output code symbols to obtain output data bits, wherein said output data bits correspond to the digital input data.
 11. The method of claim 10, wherein said step of decoding comprises the steps of:identifying with a sliding window a sequence of code symbols that represents a given code symbol set; and in a table comprising a number of entries corresponding to a number of possible unique sequences of code symbols, each entry identified by one of said possible unique sequences of code symbols, and wherein each entry identifies an output data bit set that is a decoded data bit set for the code symbol set represented by the sequence of code symbols that identifies that entry, looking up with said identified sequence of code symbols the output data bit for said given code symbol.
 12. The method according to claim 11, wherein said table provides an output bit set x' for each sequence of code symbols y' according to the following table:

    __________________________________________________________________________     Sequence of Code Symbols                                                                   x' Sequence of Code Symbols                                                                   x' Sequence of Code Symbol                                                                    x'                                   __________________________________________________________________________     000001000100                                                                               000                                                                               000020001000                                                                               100                                                                               000200010002                                                                               011                                  000001000300                                                                               000                                                                               000020003000                                                                               100                                                                               000200010003                                                                               011                                  000001000000                                                                               001                                                                               000040002000                                                                               101                                                                               000200010004                                                                               011                                  000001000001                                                                               001                                                                               000040004000                                                                               101                                                                               000200001000                                                                               100                                  000001000002                                                                               001                                                                               000040000200                                                                               110                                                                               000200003000                                                                               100                                  000001000003                                                                               001                                                                               000040000400                                                                               110                                                                               000200020000                                                                               101                                  000001000004                                                                               001                                                                               000040000100                                                                               111                                                                               000200020001                                                                               101                                  000001000010                                                                               010                                                                               000040000300                                                                               111                                                                               000200020002                                                                               101                                  000001000030                                                                               010                                                                               000100000100                                                                               000                                                                               000200020003                                                                               101                                  000001000020                                                                               010                                                                               000100000300                                                                               000                                                                               000200020004                                                                               101                                  000001000040                                                                               010                                                                               000100000200                                                                               001                                                                               000200030000                                                                               110                                  000003000100                                                                               011                                                                               000100000400                                                                               001                                                                               000200030001                                                                               110                                  000003000300                                                                               011                                                                               000100002000                                                                               010                                                                               000200030002                                                                               110                                  000001000200                                                                               100                                                                               000100004000                                                                               010                                                                               000200030003                                                                               110                                  000001000400                                                                               100                                                                               000100010000                                                                               011                                                                               000200030004                                                                               110                                  000003000000                                                                               101                                                                               000100010001                                                                               011                                                                               000200040000                                                                               111                                  000003000001                                                                               101                                                                               000100010002                                                                               011                                                                               000200040001                                                                               111                                  000003000002                                                                               101                                                                               000100010003                                                                               011                                                                               000200040002                                                                               111                                  000003000003                                                                               101                                                                               000100010004                                                                               011                                                                               000200040003                                                                               111                                  000003000004                                                                               101                                                                               000100001000                                                                               100                                                                               000200040004                                                                               111                                  000003000010                                                                               110                                                                               000100003000                                                                               100                                                                               000300000100                                                                               000                                  000003000020                                                                               110                                                                               000100020000                                                                               101                                                                               000300000300                                                                               000                                  000003000030                                                                               110                                                                               000100020001                                                                               101                                                                               000300000200                                                                               001                                  000003000040                                                                               110                                                                               000100020002                                                                               101                                                                               000300000400                                                                               001                                  000003000200                                                                               111                                                                               000100020003                                                                               101                                                                               000300002000                                                                               010                                  000003000400                                                                               111                                                                               000100020004                                                                               101                                                                               000300004000                                                                               010                                  000002000100                                                                               000                                                                               000100030000                                                                               110                                                                               000300010000                                                                               011                                  000002000300                                                                               000                                                                               000100030001                                                                               110                                                                               000300010001                                                                               011                                  000002000000                                                                               001                                                                               000100030002                                                                               110                                                                               000300010002                                                                               011                                  000002000001                                                                               001                                                                               000100030003                                                                               110                                                                               000300010003                                                                               011                                  000002000002                                                                               001                                                                               000100030004                                                                               110                                                                               000300010004                                                                               011                                  000002000003                                                                               001                                                                               000100040000                                                                               111                                                                               000300001000                                                                               100                                  000002000004                                                                               001                                                                               000100040001                                                                               111                                                                               000300003000                                                                               100                                  000002000010                                                                               010                                                                               000100040002                                                                               111                                                                               000300020000                                                                               101                                  000002000030                                                                               010                                                                               000100040003                                                                               111                                                                               000300020001                                                                               101                                  000002000020                                                                               010                                                                               000100040004                                                                               111                                                                               000300020002                                                                               101                                  000002000040                                                                               010                                                                               000010000100                                                                               000                                                                               000300020003                                                                               101                                  000004000100                                                                               011                                                                               000010000300                                                                               000                                                                               000300020004                                                                               101                                  000004000300                                                                               011                                                                               000010000200                                                                               001                                                                               000300030000                                                                               110                                  000002000200                                                                               100                                                                               000010000400                                                                               001                                                                               000300030001                                                                               110                                  000002000400                                                                               100                                                                               000010002000                                                                               010                                                                               000300030002                                                                               110                                  000004000000                                                                               101                                                                               000010004000                                                                               010                                                                               000300030003                                                                               110                                  000004000001                                                                               101                                                                               000030001000                                                                               011                                                                               000300030004                                                                               110                                  000004000002                                                                               101                                                                               000030003000                                                                               011                                                                               000300040000                                                                               111                                  000004000003                                                                               101                                                                               000010001000                                                                               100                                                                               000300040001                                                                               111                                  000004000004                                                                               101                                                                               000010003000                                                                               100                                                                               000300040002                                                                               111                                  000004000010                                                                               110                                                                               000030002000                                                                               101                                                                               000300040003                                                                               111                                  000004000030                                                                               110                                                                               000030004000                                                                               101                                                                               000300040004                                                                               111                                  000004000020                                                                               110                                                                               000030000200                                                                               110                                                                               000400000100                                                                               000                                  000004000040                                                                               110                                                                               000030000400                                                                               100                                                                               000400000300                                                                               000                                  000004000200                                                                               111                                                                               000030000100                                                                               111                                                                               000400000200                                                                               001                                  000004000400                                                                               111                                                                               000030000300                                                                               111                                                                               000400000400                                                                               001                                  000020000100                                                                               000                                                                               000200000100                                                                               000                                                                               000400002000                                                                               010                                  000020000300                                                                               000                                                                               000200000300                                                                               000                                                                               000400004000                                                                               010                                  000020000200                                                                               001                                                                               000200000200                                                                               001                                                                               000400000000                                                                               011                                  000020000400                                                                               001                                                                               000200000400                                                                               001                                                                               000400010001                                                                               011                                  000020002000                                                                               010                                                                               000200002000                                                                               010                                                                               000400010002                                                                               011                                  000020004000                                                                               010                                                                               000200004000                                                                               010                                                                               000400010003                                                                               011                                  000040001000                                                                               011                                                                               000200010000                                                                               011                                                                               000400010004                                                                               011                                  000040003000                                                                               011                                                                               000200010001                                                                               011                                                                               000400001000                                                                               100                                  000400003000                                                                               100                                                                               001000200002                                                                               100                                                                               003000200002                                                                               100                                  000400020000                                                                               101                                                                               001000200003                                                                               100                                                                               003000200003                                                                               100                                  000400020001                                                                               101                                                                               001000200004                                                                               100                                                                               003000200004                                                                               100                                  000400020002                                                                               101                                                                               001000200010                                                                               101                                                                               003000200010                                                                               101                                  000400020003                                                                               101                                                                               001000200030                                                                               101                                                                               003000200030                                                                               101                                  000400020004                                                                               101                                                                               001000200020                                                                               101                                                                               003000200020                                                                               101                                  000400030000                                                                               110                                                                               001000200040                                                                               101                                                                               003000200040                                                                               101                                  000400030001                                                                               110                                                                               001000400000                                                                               110                                                                               003000400000                                                                               110                                  000400030002                                                                               110                                                                               001000400001                                                                               110                                                                               003000400001                                                                               110                                  000400030003                                                                               110                                                                               001000400002                                                                               110                                                                               003000400002                                                                               110                                  000400030004                                                                               110                                                                               001000400003                                                                               110                                                                               003000400003                                                                               110                                  000400040000                                                                               111                                                                               001000400004                                                                               110                                                                               003000400004                                                                               110                                  000400040001                                                                               111                                                                               001000400010                                                                               111                                                                               003000400010                                                                               111                                  000400040002                                                                               111                                                                               001000400030                                                                               111                                                                               003000400030                                                                               111                                  000400040003                                                                               111                                                                               001000400020                                                                               111                                                                               003000400020                                                                               111                                  000400040004                                                                               111                                                                               001000400040                                                                               111                                                                               003000400040                                                                               111                                  001000000100                                                                               000                                                                               003000000100                                                                               000                                                                               002000000100                                                                               000                                  001000000300                                                                               000                                                                               003000000300                                                                               000                                                                               002000000300                                                                               000                                  001000000200                                                                               001                                                                               003000000200                                                                               001                                                                               002000000200                                                                               001                                  001000000400                                                                               001                                                                               003000000400                                                                               001                                                                               002000000400                                                                               001                                  001000002000                                                                               010                                                                               003000002000                                                                               010                                                                               002000002000                                                                               010                                  001000004000                                                                               010                                                                               003000004000                                                                               010                                                                               002000004000                                                                               010                                  001000010000                                                                               011                                                                               003000010000                                                                               011                                                                               002000010000                                                                               011                                  001000010001                                                                               011                                                                               003000010001                                                                               011                                                                               002000010001                                                                               011                                  001000010002                                                                               011                                                                               003000010002                                                                               011                                                                               002000010002                                                                               011                                  001000010003                                                                               011                                                                               003000010003                                                                               011                                                                               002000010003                                                                               011                                  001000010004                                                                               011                                                                               003000010004                                                                               011                                                                               002000010004                                                                               011                                  001000001000                                                                               100                                                                               003000001000                                                                               100                                                                               002000001000                                                                               100                                  001000003000                                                                               100                                                                               003000003000                                                                               100                                                                               002000003000                                                                               100                                  001000020000                                                                               101                                                                               003000020000                                                                               101                                                                               002000020000                                                                               101                                  001000020001                                                                               101                                                                               003000020001                                                                               101                                                                               002000020001                                                                               101                                  001000020002                                                                               101                                                                               003000020002                                                                               101                                                                               002000020002                                                                               101                                  001000020003                                                                               101                                                                               003000020003                                                                               101                                                                               002000020003                                                                               101                                  001000020004                                                                               101                                                                               003000020004                                                                               101                                                                               002000020004                                                                               101                                  001000030000                                                                               110                                                                               003000030000                                                                               110                                                                               002000030000                                                                               110                                  001000030001                                                                               110                                                                               003000030001                                                                               110                                                                               002000030001                                                                               110                                  001000030002                                                                               110                                                                               003000030002                                                                               110                                                                               002000030002                                                                               110                                  001000030003                                                                               110                                                                               003000030003                                                                               110                                                                               002000030003                                                                               110                                  001000030004                                                                               110                                                                               003000030004                                                                               110                                                                               002000030004                                                                               110                                  001000040000                                                                               111                                                                               003000040000                                                                               111                                                                               002000040000                                                                               111                                  001000040001                                                                               111                                                                               003000040001                                                                               111                                                                               002000040001                                                                               111                                  001000040002                                                                               111                                                                               003000040002                                                                               111                                                                               002000040002                                                                               111                                  001000040003                                                                               111                                                                               003000040003                                                                               111                                                                               002000040003                                                                               111                                  001000040004                                                                               111                                                                               003000040004                                                                               111                                                                               002000040004                                                                               111                                  001000100000                                                                               000                                                                               003000100000                                                                               000                                                                               002000100000                                                                               000                                  001000100001                                                                               000                                                                               003000100001                                                                               000                                                                               002000100001                                                                               000                                  001000100002                                                                               000                                                                               003000100002                                                                               000                                                                               002000100002                                                                               000                                  001000100003                                                                               000                                                                               003000100003                                                                               000                                                                               002000100003                                                                               000                                  001000100004                                                                               000                                                                               003000100004                                                                               000                                                                               002000100004                                                                               000                                  001000100010                                                                               001                                                                               003000100010                                                                               001                                                                               002000100010                                                                               001                                  001000100030                                                                               001                                                                               003000100030                                                                               001                                                                               002000100030                                                                               001                                  001000100020                                                                               001                                                                               003000100020                                                                               001                                                                               002000100020                                                                               001                                  001000100040                                                                               001                                                                               003000100040                                                                               000                                                                               002000100040                                                                               001                                  001000300000                                                                               010                                                                               003000300000                                                                               010                                                                               002000300000                                                                               010                                  001000300001                                                                               010                                                                               003000300001                                                                               010                                                                               002000300001                                                                               010                                  001000300002                                                                               010                                                                               003000300002                                                                               010                                                                               002000300002                                                                               010                                  001000300003                                                                               010                                                                               003000300003                                                                               010                                                                               002000300003                                                                               010                                  001000300004                                                                               010                                                                               003000300004                                                                               010                                                                               002000300004                                                                               010                                  001000300010                                                                               011                                                                               003000300010                                                                               011                                                                               002000300010                                                                               011                                  001000300030                                                                               011                                                                               003000300030                                                                               011                                                                               002000300030                                                                               011                                  001000300020                                                                               011                                                                               003000300020                                                                               011                                                                               002000300020                                                                               011                                  001000300040                                                                               011                                                                               003000300040                                                                               011                                                                               002000300040                                                                               011                                  001000200000                                                                               100                                                                               003000200000                                                                               100                                                                               002000200000                                                                               100                                  001000200001                                                                               100                                                                               003000200001                                                                               100                                                                               002000200001                                                                               100                                  002000200002                                                                               100                                                                               004000200000                                                                               100                                                                               010000030000                                                                               110                                  002000200003                                                                               100                                                                               004000200001                                                                               100                                                                               010000030001                                                                               110                                  002000200004                                                                               100                                                                               004000200002                                                                               100                                                                               010000030002                                                                               110                                  002000200010                                                                               101                                                                               004000200003                                                                               100                                                                               010000030003                                                                               110                                  002000200030                                                                               101                                                                               004000200004                                                                               100                                                                               010000030004                                                                               110                                  002000200020                                                                               101                                                                               004000200010                                                                               101                                                                               010000040000                                                                               111                                  002000200040                                                                               101                                                                               004000200030                                                                               101                                                                               010000040001                                                                               111                                  002000400000                                                                               110                                                                               004000200020                                                                               101                                                                               010000040002                                                                               111                                  002000400001                                                                               110                                                                               004000200040                                                                               101                                                                               010000040003                                                                               111                                  002000400002                                                                               110                                                                               004000400000                                                                               110                                                                               010000040004                                                                               111                                  002000400003                                                                               110                                                                               004000400000                                                                               110                                                                               010000100000                                                                               000                                  002000400004                                                                               110                                                                               004000400002                                                                               110                                                                               010000100001                                                                               000                                  002000400010                                                                               111                                                                               004000400003                                                                               110                                                                               010000100002                                                                               000                                  002000400030                                                                               111                                                                               004000400004                                                                               110                                                                               010000100003                                                                               000                                  002000400020                                                                               111                                                                               004000400010                                                                               111                                                                               010000100004                                                                               000                                  002000400040                                                                               111                                                                               004000400030                                                                               111                                                                               010000100010                                                                               001                                  004000000100                                                                               000                                                                               004000400020                                                                               111                                                                               010000100030                                                                               001                                  004000000300                                                                               000                                                                               004000400040                                                                               111                                                                               010000100020                                                                               001                                  004000000200                                                                               001                                                                               010001000100                                                                               000                                                                               010000100040                                                                               001                                  004000000400                                                                               001                                                                               010001000300                                                                               000                                                                               010000300000                                                                               010                                  004000002000                                                                               010                                                                               010001000000                                                                               001                                                                               010000300001                                                                               010                                  004000004000                                                                               010                                                                               010001000001                                                                               001                                                                               010000300002                                                                               010                                  004000010000                                                                               011                                                                               010001000002                                                                               001                                                                               010000300003                                                                               010                                  004000010001                                                                               011                                                                               010001000003                                                                               001                                                                               010000300004                                                                               010                                  004000010002                                                                               011                                                                               010001000004                                                                               001                                                                               010000300010                                                                               011                                  004000010003                                                                               011                                                                               010001000010                                                                               010                                                                               010000300030                                                                               011                                  004000010004                                                                               011                                                                               010001000030                                                                               010                                                                               010000300020                                                                               011                                  004000001000                                                                               100                                                                               010001000020                                                                               010                                                                               010000300040                                                                               011                                  004000003000                                                                               100                                                                               010001000040                                                                               010                                                                               010000200000                                                                               100                                  004000020000                                                                               101                                                                               010003000100                                                                               011                                                                               010000200001                                                                               100                                  004000020001                                                                               101                                                                               010003000300                                                                               011                                                                               010000200002                                                                               100                                  004000020002                                                                               101            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110                                                                               100002000002                                                                               001                                  400030003000                                                                               011                                                                               400004000030                                                                               110                                                                               100002000003                                                                               001                                  100002000004                                                                               001                                                                               300030002000                                                                               101                                                  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                                                    100                                                                               300004000040                                                                               110                                  100004000020                                                                               110                                                                               100030002000                                                                               101                                                                               300004000200                                                                               111                                  100004000040                                                                               110                                                                               100030004000                                                                               101                                                                               300004000400                                                                               111                                  100004000200                                                                               111                                                                               100030000200                                                                               110                                                                               300001000100                                                                               000                                  100004000400                                                                               111                                                                               100030000400                                                                               110                                                  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                                      100                                                                               300040001000                                                                               011                                                                               300001000040                                                                               010                                  100020003000                                                                               100                                                                               300040003000                                                                               011                                                                               300003000100                                                                               011                                  100040002000                                                                               101            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111                                                                               300003000003                                                                               101                                  300010000300                                                                               000                                                                               300040000300                                                                               111                                                                               300003000004                                                                               101                                  300010000200                                                                               001                                                                               300002000100                                                                               000                                                                               300003000010                                                                               110                                  300010000400                                                                               001                                                                               300002000300                                                                               000                                                                               300003000030                                                                               110                                  300010002000                                                                               010                                                                               300002000000                                                                               001                                                                               300003000020            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